Problem: Let $f$ be a scalar field. Is the expression $\nabla(\nabla f)$ a scalar field, a vector field, or undefined? Choose 1 answer: Choose 1 answer: (Choice A) A Scalar field (Choice B) B Vector field (Choice C) C Undefined
Answer: The gradient, which takes a scalar field and gives a vector field, can be written in two ways: $\text{grad}(f) = \nabla f$ Therefore, $\nabla(\nabla f)$ is the gradient of the gradient of a scalar field. The gradient of a scalar field is a vector field. Because the gradient only takes scalar fields, the gradient of that vector field is undefined. The expression $\nabla(\nabla f)$ is undefined.